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Cyclotomic root systems and bad primes
Michel Broué, Ruth Corran, Jean Michel
To cite this version:
Michel Broué, Ruth Corran, Jean Michel. Cyclotomic root systems and bad primes. 2017. �hal-01506943v2�
MICHEL BROU´E, RUTH CORRAN, JEAN MICHEL Abstract. We generalize the definition and properties of root systems to complex reflection groups — roots become rank one projective modules over the ring of integers of a number field k.
In the irreducible case, we provide a classification of root systems over the field of definition k of the reflection representation.
In the case of spetsial reflection groups, we generalize as well the definition and properties of bad primes.
Contents
1. Introduction 2
2. Complex reflection groups 3
2.1. Preliminary material about reflections 4
2.2. Reflection groups 6
2.3. The Shephard–Todd classification 11
2.4. Parabolic subgroups 12
2.5. Linear characters of a finite reflection group 13
3. Root Systems 14
3.1. Zk-roots 15
3.2. Zk-root systems 17
3.3. Root systems and parabolic subgroups 24
3.4. Root lattices, root bases 25
3.5. Example: the Weyl group of type B2 29
3.6. Connection index 30
4. The cyclic groups 32
4.1. Generalities 32
4.2. The case of G = µ2 33
4.3. A few particular cases for G = µ3 34
5. The group G(de, e, r) 36
5.1. The reflections of G(de, e, r). 36
5.2. The complete reduced Zk-root system R(de, e, r). 36 5.3. Classifying complete reduced root systems for G(de, e, r) 38
5.4. Root systems for G(e, e, 2) on its field of definition 40
Date: April 13, 2017.
1991 Mathematics Subject Classification. Primary : 20F55 ; Secondary: 20C08. Key words and phrases. Complex reflection groups, root systems, cyclotomic Hecke algebras, spetses.
We thank Gabriele Nebe for her previous seminal work on this topic. 1
5.5. Classifying distinguished root systems for G(de, e, r) 41
6. Principal root systems and Cartan matrices 43
6.1. Cartan matrices 43
6.2. Free root lattices 45
6.3. Principal root bases and Cartan matrices for imprimitive well generated reflection groups 6.4. The case of not well-generated groups G(de, e, r) 48
7. Reflection groups and root systems over R 53
7.1. Preliminary: positive Hermitian forms 53
7.2. Families of simple reflections for real reflection groups 54
7.3. Highest half-lines 59
7.4. Real root systems 60
8. Bad numbers 61
9. Classification of distinguished root systems for irreducible primitive reflection groups 66
9.1. Cases with only one genus of distinguished root systems 67
9.2. The cases with more than one genus 68
Appendix A. On roots of unity 71
A.1. Notation and summary of known properties 71
A.2. Decomposition of the ideal Im,n in Zn 72
A.3. On cyclotomic fields 74
Appendix B. A table of Cartan matrices 77
References 81
Index 82
1. Introduction
The spirit of the Spetses program ([BMM], [BMM2]) is to consider (at least some of) the complex reflections groups as Weyl groups for some mysterious object which looks like a “generic finite reductive group” — and which is yet unknown.
Some of the data attached to finite reductive groups, such as the parameterization of unipotent characters, their generic degrees, Frobe-nius eigenvalues, and also the families and their Fourier matrices, turn out to depend only on the Q-representation of the Weyl group. But supplementary data, such as the parameterization of unipotent classes, the values of unipotent characters on unipotent elements, depend on the entire root datum.
For a complex reflection group, not necessarily defined over Q, but defined over a number field k, it thus seems both necessary and natural to study “Zk-root data”, where Zk denotes the ring of integers of k.
Some related work has already been done in that direction, in par-ticular by Nebe [Ne], who classified Zk-lattices invariant under the
re-flection group – and whose work inspired us. More recently, motivated by some work on p-compact groups, Grodal and others [AnGr], [Gr] defined root data for reflection groups defined over finite fields.
Here we define and classify Zk-root systems1, as well as root lattices
and coroot lattices, for all complex reflection groups. Of course most of the rings Zk are not principal ideal domains (although — quite a
remarkable fact — they are P.I.D. for the 34 exceptional irreducible complex reflection groups) and one has naturally to replace elements of Zk by ideals of Zk. Taking this into account, our definition of root
system mimics Bourbaki’s definition [BouLie] and of course working with “ideal numbers” is much more appropriate, on general Dedekind domains, than working with “numbers”, as suggested by what follows. • A complex reflection group may occur as a parabolic subgroup of another reflection group whose field of definition is larger. A first problem with considering vectors (as in the usual ap-proach of root systems) instead of one-dimensional Zk-modules
(as we do here) is that we would have “too many” of them when restricting to a parabolic subgroup.
• In the case of the group of type B2, over a field where the ideal
generated by 2 has a square root, such as Q(√2) or Q(√−1), not only do we find the usual system of type B2 but we also find a
system which affords the exterior automorphism2B
2. If we were
considering numbers instead of ideals, this automorphism would exist only if the number 2 (rather than the ideal generated by 2) has a square root.
The exceptional group denoted by G29, defined over Q(√−1),
has a subgroup of type B2 and the normaliser of this subgroup
induces the automorphism 2B
2. Our corresponding root system
has the same automorphism since 1 + i and 1− i generate the same ideal and (1 + i)(1− i) = 2, thus 2 has an “ideal square root” in the ring Z[√−1].
Perhaps the most interesting and intriguing fact which comes out of the classification concerns the generalisation of the notions of connec-tion index and bad primes: in the case of spetsial reflecconnec-tion groups, the order of the group is divisible by the factorial of the rank times the connection index, and the bad primes for the corresponding Spets make up the remainder, just as in the case of finite reductive groups and Weyl groups.2
2. Complex reflection groups
We denote λ 7→ λ∗ the complex conjugation and we denote by k a
subfield of C stable by complex conjugation.
1By analogy with the well established terminology “cyclotomic Hecke algebras”
— a crucial notion in the Spetses program —, we propose to call these generalised root systems “cyclotomic root systems”.
2Notice though that, as shown by Nebe [Ne], the bad primes for spetsial groups
(see Section 8) do not occur as divisors of the orders of the quotient of the root lattice by the root lattice of maximal reflection subgroups.
2.1. Preliminary material about reflections.
Let (V, W ) be a pair of finite dimensional k-vector spaces with a given Hermitian pairing V × W → k : (v, w) 7→ hv, wi; that is:
• h-, -i is linear in V and semi-linear in W : for λ, µ ∈ k, v ∈ V and w∈ W we have hλv, µwi = λµ∗hv, wi.
• hv, wi = 0 for all w ∈ W implies v = 0. • hv, wi = 0 for all v ∈ V implies w = 0.
Similarly (w, v)7→ hv, wi∗ defines a Hermitian pairing W×V → k that we will also denote (w, v)7→ hw, vi when its meaning is clear from the context.
Any vector space can be naturally endowed with a Hermitian pairing with its twisted dual:
Definition 2.1. The twisted dual of a k-vector space V , denoted ∗V ,
is the k-vector space which is the conjugate under ∗ of the dual V∗ of
V . In other words,
• as an abelian group, ∗V = V∗,
• an element λ ∈ k acts on ∗V as λ∗ acts on V∗.
The pairing V ×∗V → k : (v, φ) 7→ φ(v) is a Hermitian pairing, called
the canonical pairing associated with V .
When V is a real vector space, the twisted dual is the usual dual. Let G(V,W ) be the subgroup of GL(V ) × GL(W ) which preserves
the pairing. The first (resp. second) projection gives an isomorphism G(V,W )−→ GL(V ) (resp. G∼ (V,W )−→ GL(W )). Composing the second∼
isomorphism with the inverse of the first we get an isomorphism g 7→ g∨ : GL(V )−→ GL(W ). The inverse morphism has the same definition∼
reversing the roles of V and W , and we will still denote it g 7→ g∨ so
that (g∨)∨ = g.
In the case of the canonical pairing associated with V , the isomor-phism g 7→ g∨ is just the contragredient tg−1.
Definition 2.2. A reflection is an element s ∈ GL(V ) of finite order such that ker(s− 1) is an hyperplane. Define the
• reflecting hyperplane of s as Hs := ker(s− 1),
• reflecting line of s as Ls := im (s− 1),
• dual reflecting line Ms of s as the orthogonal (in W ) of Hs,
• dual reflecting hyperplane Ks of s as the orthogonal (in W ) of
Ls.
Denote by ζs the determinant of s, which is a root of unity.
It is clear that a reflection s is determined by Hs, Ls and ζs. In turn,
Hs is determined by the dual reflecting line Ms. Note that Ms is not
orthogonal to Lssince Hsdoes not contain Ls. Thus giving a reflection
is equivalent to giving the following data:
• L is a line in V and M a line in W which are not orthogonal. • ζ ∈ k× is a root of unity.
Formulae for the reflections defined by a reflection triple are sym-metric in V and W :
Proposition 2.4. A reflection triple (L, M, ζ) defines a pair of reflec-tions (s, s∨) ∈ GL(V ) × GL(W ) (which preserve the pairing) by the
formulae:
s(v) = v− hv, yi
hx, yi(1− ζ)x, s∨(w) = w−hw, xi
hy, xi(1− ζ)y, for any non-zero x∈ L and y ∈ M.
Proof. An easy computation shows that the pair (s, s∨) of reflections
preserves the pairing h-, -i. Furthermore, the reflection s defined by this formula determines the triple (L, M, ζ), and the reflection s∨ (in
W ) determines the triple (M, L, ζ).
As long as ζm 6= 1, the pair (sm, s∨m) is precisely the pair of
reflec-tions defined by the triple (L, M, ζm): the order of s is the order of the
element ζ ∈ k×.
To summarize, we have:
Proposition 2.5. Reflection triples are in bijection with • reflections in GL(V ),
• reflections in GL(W ),
• pairs of reflections (s, s∨) in GL(V )× GL(W ) which preserve
the pairing h-, -i.
An element g ∈ GL(V ) acts naturally on reflection triples (L, M, ζ) through the action of (g, g∨) on pairs (L, M). It follows from the
pre-vious proposition that g commutes with a reflection s if and only if (g, g∨) stabilizes (L
s, Ms).
Notation 2.6. Let t = (L, M, ζ) be a reflection triple; denote by st the
corresponding reflection, and write s∨
t for the reflection corresponding
to (M, L, ζ). We will also write Lt, Ht, Mt, Kt for Lst, Hst, Mst, Kst.
Stable subspaces.
A reflection is diagonalisable, hence so is its restriction to a stable subspace. The next lemma follows directly.
Lemma 2.7. Let V1 be a subspace of V stable by a reflection s. Then
• either V1 is fixed by s (i.e., V1 ⊆ Hs),
• or V1 contains Ls, and then V1 = Ls⊕ (Hs∩ V1), in which case
the restriction of s to V1 is a reflection.
In particular, the restriction of a reflection to a stable subspace is either trivial or a reflection.
Commuting reflections.
Lemma 2.8. Let t1 = (L1, M1, ζ1) and t2 = (L2, M2, ζ2) be two
reflec-tion triples. We have the following three sets of equivalent asserreflec-tions.
(I) (i) (st1, s∨t1) acts trivially on (L2, M2).
(ii) (st2, s∨t2) acts trivially on (L1, M1).
(iii) L1 ⊆ Ht2 and L2 ⊆ Ht1, in which case we say that t1 and
t2 are orthogonal.
(II) (i) (st1, s∨t1) acts by ζ1 on (L2, M2).
(ii) (st2, s∨t2) acts by ζ2 on (L1, M1).
(iii) L1 = L2 and Ht1 = Ht2, in which case we say that t1 and
t2 are parallel.
(III) (i) st1st2 = st2st1,
(ii) st1 stabilizes t2.
(iii) st2 stabilizes t1.
(iv) t1 and t2 are either orthogonal or parallel.
Proof. The proof of both (I) and (II) proceeds by showing the equiva-lence of (i) and (iii), from which the equivaequiva-lence between (ii) and (iii) follows by symmetry. (III) then follows from (I), (II), and the
defini-tions.
From now on and until the end of this section, we shall work “on the V ” side. Nevertheless, we shall go on using notions previously introduced in connection with an Hermitian pairing V × W → k. 2.2. Reflection groups.
Definition 2.9. A reflection group on k is a pair (V, G), where V is a finite dimensional k-vector space and G is a subgroup of GL(V ) generated by reflections. A reflection group is said to be finite if G is finite, and complex if k ⊆ C.
Throughout this subsection, (V, G) denotes a reflection group. Whenever a set of reflections S generates G, then Ts∈SHs = VG,
the set of elements fixed by G.
Definition 2.10. A reflection group (V, G) is essential if VG={0}.
Definition 2.11. A set of reflections is saturated if it is closed under conjugation by the group it generates.
When S is saturated, G is a (normal) subgroup of the subgroup of GL(V ) which stabilizes S, and the subspace VS, defined by
VS :=
X
s∈S
Ls,
Orthogonal decomposition. Definition 2.12.
(1) Define an equivalence relation ∼ on S as the transitive closure of: st∼ st′ whenever t is not orthogonal to t′.
(2) The set of reflections S is said to be irreducible if it consists of a unique ∼-equivalence class.
Lemma 2.13. Let S be a saturated set of reflections which generates the reflection group (V, G). Then the ∼-equivalence classes of S are stable under G-conjugacy.
Proof. This results from the stability under G-conjugacy of S and of
the relation “being orthogonal”.
When a group G acts on a set V (which could be G itself on which G acts by conjugation), for X ⊂ V we denote by NG(X) the normalizer
(stabilizer) of X in G, i.e., the set of g ∈ G such that g(X) = X. We denote by CG(X) the centralizer (fixator) of X, i.e., the set of g ∈ G
such that, for all x∈ X, g(x) = x. Notice that CG(X) ⊳ NG(X).
Lemma 2.14. Let S be a set of reflections on V .
(1) The action of NGL(V )(S) on S induces an injection:
NGL(V )(S)/CGL(V )(S) ֒→ S(S),
into the symmetric group on S.
(2) If S is saturated and irreducible, then CGL(V )(S) acts by scalars
on VS.
Proof. Item (1) is trivial. Let us prove (2).
Take any g ∈ CGL(V )(S). The pair (g, g∨) stabilizes the reflection
triples of all reflections in S. Take two non-orthogonal reflections s, s′in
S with corresponding reflection triples (L, M, ζ), (L′, M′, ζ′) and choose
non-zero elements x ∈ L, y ∈ M, v ∈ L′, w ∈ M′. Then g(x) = λx and
g(v) = µv for some non-zero scalars λ and µ.
Since S is saturated, it also contains the reflection corresponding to the triple s · (L′, M′, ζ′). But s(v) = v− hv,yi
hx,yi(1− ζ)x, so g(s(v)) =
µv− hv,yihx,yi(1− ζ)λx = αs(v) for some non-zero scalar α. In particular, this implies µ = λ, and g acts by scalar multiplication on Ls+ Ls′.
Since S is irreducible, this shows that g acts by scalar multiplication on Ls+ Lt for any pair s, t∈ S, hence acts by scalar multiplication on
VS.
Lemma 2.15. The number of ∼-equivalence classes of reflections in S is bounded by the dimension of V . In particular, it is finite.
Proof. Assume that t1 = (L1, M1, ζ1) , . . . , tm = (Lm, Mm, ζm)
corre-spond to reflections which belong to distinct ∼-equivalence classes. So in particular all the ti are mutually orthogonal. This implies that the
Li are linearly independent: for let xi ∈ Li, yi ∈ Mi be non-zero
el-ements and assume λ1x1 + λ2x2 +· · · + λmxm = 0. The Hermitian
product with yi yields λihxi, yii = 0, hence λi = 0.
The following lemma is straightforward to verify.
Lemma 2.16. Let S be a set of reflections which generate the reflection group (V, G). Let S = S1 ⊔ S2 ⊔ · · · ⊔ Sm be the decomposition of S
into ∼-equivalence classes. Denote by Gi the subgroup of G generated
by the reflections s for s ∈ Si and by Vi the subspace of V generated by
the lines Ls for s∈ Si.
(1) The group Gi acts trivially on Pj6=iVj.
(2) For 1 ≤ i 6= j ≤ m, the groups Gi and Gj commute.
(3) G = G1G2. . . Gm.
The above result can be refined in the case where G acts completely reducibly on V – then the decomposition is direct (see 2.19 below).
Lemma 2.16 can be applied to obtain a criterion for finiteness of G. Proposition 2.17. Let G be the group generated by a saturated set of reflections S. If S is finite, then G is finite.
Proof. Consider the case where S is irreducible. By the second part of Lemma2.14, we know that the centralizer of S is contained in k×, so in
particular, G∩ CGL(V )(S) is contained in k×. Moreover, each reflection
has finite order, so in fact G∩ CGL(V )(S) ⊂ µ(k), the set of roots of
unity of k. Now the determinants of the elements of G belong to the finite subgroup of µ(k) generated by the determinants of the elements of S. Thus G∩ CGL(V )(S) is finite.
As G is contained in NGL(V )(S), by the first part of Lemma 2.14, it
is an extension of G∩ CGL(V )(S) by a subgroup of S(S). By finiteness
of S, this is a finite extension. Hence G is finite.
Finally, consider the general case G = G1· · · Gm, where the groups
Gi are generated by reflections in distinct equivalence classes, as in
Lemma 2.16. By the preceding comments, each Gi is finite; so G must
also be finite.
The case when G is completely reducible.
Proposition 2.18. Let S be a set of reflections generating the reflec-tion group (V, G), and suppose that the acreflec-tion of G on V is completely reducible. Then
(1) V = VS⊕ VG, and
(2) the restriction from V to VS induces an isomorphism from G
onto its image in GL(VS), an essential reflection group on VS.
Proof. The subspace VS is G-stable, hence since G is completely
have VS ⊃ Ls for all s ∈ S, thus (by Lemma 2.7) V′ is contained in
Hs; it follows that V′ ⊆ Ts∈SHs ⊆ VG. So it suffices to prove that
VS∩ VG = 0.
Since VG is stable by G, there exists a complementary subspace V′′
which is stable by G. Whenever s ∈ S, we have Ls ⊆ V′′ (otherwise,
by Lemma 2.7, we have V′′ ⊆ H
r, which implies that s is trivial since
V = VG ⊕ V′′, a contradiction). This shows that V
S ⊆ V′′, and in
particular that VS∩ VG = 0.
Proposition 2.19. Let S be a set of reflections generating the reflec-tion group (V, G), and suppose that the acreflec-tion of G on V is completely reducible. Denote by {Vi}i the subspaces associated to an orthogonal
decomposition as in Lemma 2.16.
(1) For 1 ≤ i ≤ m, the action of Gi on Vi is irreducible.
(2) VS =Li=mi=1 Vi.
(3) G = G1× G2× · · · × Gm
Proof. (1) The subspace Vi is stable under G, and the action of G on a
stable subspace is completely reducible. But the image of G in GL(Vi)
is the same as the image of Gi. So the action of Gi on Vi is completely
reducible.
So we write Vi = Vi′⊕ Vi′′ where Vi′ and Vi′′ are stable by Gi. Define:
Si′ :={s ∈ Si| Ls⊆ Vi′} and Si′′:={s ∈ Si| Ls⊆ Vi′′} .
By Lemma 2.7, if s∈ S′
i, then Vi′′ ⊆ Hs, and if s ∈ Si′′, then Vi′ ⊆ Hs.
Hence any two elements of S′
i and Si′′ are mutually orthogonal. Thus
one of them has to be all of Si.
(2) By Lemma 2.16, we have Pj6=iVj ⊂ VGi. By (1), and by
Propo-sition 2.18, we then get Vi∩Pj6=iVj = 0 .
(3) An element g ∈ Gi which also belongs to Qj6=iGj acts trivially
on Vi. Since (by (1) and by Proposition 2.18) the representation of Gi
on Vi is faithful, we see that g = 1.
Reflecting pairs.
For H a reflecting hyperplane, notice that
CG(H) ={1} ∪ {g ∈ G | ker (g − 1) = H} .
For L a reflecting line, we have:
CG(V /L) ={1} ∪ {g ∈ G | im (g − 1) = L} .
So CG(V /L) is the group of all elements of G which stabilize L and
which act trivially on V /L; a normal subgroup of NG(L).
Similarly, if M is a dual reflecting line in W , CG(W/M) ⊳ NG(M).
Remark 2.20. Recall that orthogonality between V and W induces a bijection between reflecting hyperplanes and dual reflecting lines, as well as between reflecting lines and dual reflecting hyperplanes.
Then if M is the orthogonal (in W ) of the reflecting hyperplane H, we have
CG(H) = CG(W/M) .
We will be considering the following property.
Property 2.21. The reflection group (V, G) is such that the represen-tations of G and all its proper subgroups on V are completely reducible.
Note that this is the case in particular when G is finite.
Proposition 2.22. Let (V, G) be a reflection group with Property 2.21. (1) Let H be a reflecting hyperplane for G. There exists a unique
reflecting line L such that CG(V /L) = CG(H).
In other words:
Let M be a dual reflecting line for G. There exists a unique reflecting line L such that CG(V /L) = CG(W/M).
(2) Let L be a reflecting line for G. There exists a unique reflecting hyperplane H such that CG(H) = CG(V /L).
In other words: Let K be a dual reflecting hyperplane for G. There exists a unique reflecting hyperplane H such that CG(K) = CG(H).
(3) If (L, H) (or (L, M), or (H, K)) is a pair as above, then (a) CG(H) consists of the identity and of reflections s where
Hs = H and Ls= L,
(b) CG(H) is isomorphic to a subgroup of k×, and so is cyclic
if G is finite, and
(c) NG(H) = NG(L) = NG(M) = NG(K).
Proof of 2.22.
• Assume CG(H) 6= {1}. Since the action of CG(H) on V is
com-pletely reducible, there is a line L which is stable by CG(H) and such
that H⊕L = V . Such a line is obviously the eigenspace (corresponding to an eigenvalue different from 1) for any non-trivial element of CG(H).
This shows that L is uniquely determined, and that CG(H) consists of
1 and of reflections with hyperplane H and line L. It follows also that CG(H)⊆ G(V/L). Notice that H and L are the isotypic components
of V under the action of CG(H).
• Assume CG(V /L) 6= {1}. Since the action of CG(V /L) on V
is completely reducible, there is a hyperplane H which is stable by CG(V /L) and such that L⊕ H = V . Such a hyperplane is clearly the
kernel of any nontrivial element of CG(V /L). This shows that H is
uniquely determined, and that CG(V /L)⊆ CG(H).
We let the reader conclude the proof.
Notice the following improvement to Lemma 2.8 due to complete reducibility.
Proposition 2.23. Let t, t′ be two reflection triples such that s t and
st′ belong to a group satisfying Property 2.21. Then
(1) t and t′ are orthogonal if L
t ⊆ Ht′ or Lt′ ⊆ Ht.
(2) t and t′ are parallel if L
t= Lt′ or Ht = Ht′,
2.3. The Shephard–Todd classification.
Definition 2.24. Given (V, G) and (V′, G′) finite reflection groups on
k, an isomorphism from (V, G) to (V′, G′) is a k-linear isomorphism
f : V −→ V∼ ′ which conjugates the group G onto the group G′.
From now on in this subsection we assume that k ⊆ C.
The family of finite complex reflection groups denoted G(de, e, r) . Let d, e and r be three positive integers.
Let Dr(de) be the set of diagonal complex matrices with diagonal
entries in the group µde of all de–th roots of unity. The d–th power of the determinant defines a surjective morphism
detd: Dr(de) ։ µe.
Let A(de, e, r) be the kernel of the above morphism. In particular we have |A(de, e, r)| = (de)r/e . Identifying the symmetric group S
r with
the usual r× r permutation matrices, we define G(de, e, r) := A(de, e, r) ⋊ Sr.
We have |G(de, e, r)| = (de)rr!/e , and G(de, e, r) is the group of
all monomial r × r matrices, with entries in µde, and product of all non-zero entries in µd.
Examples 2.25.
• G(e, e, 2) is the dihedral group of order 2e.
• G(d, 1, r) is isomorphic to the wreath product µd≀Sr. For d = 2,
it is isomorphic to the Weyl group of type Br (or Cr).
• G(2, 2, r) is isomorphic to the Weyl group of type Dr.
The following theorem, stated in terms of abstract groups, is the main result of [ShTo]. It is explicitly proved in [Co, 2.4, 3.4 and 5.12]. Theorem 2.26 (Shephard–Todd)). Let (V, G) be a finite irreducible complex reflection group. Then one of the following assertions is true: • (V, G) ≃ (Cr, G(de, e, r)) for some integers d, e, r, with de≥ 2,
r≥ 1
• (V, G) ≃ (Cr−1, S
r) for some an integer r≥ 1
• (V, G) is isomorphic to one of 34 exceptional reflection groups. The exceptional groups are traditionally denoted G4, . . . , G37.
Remark 2.27. Conversely, any group G(de, e, r) is irreducible on Cr
Remark 2.28. Theorem 2.26 has the following consequence.
Assume that (V, G) is a complex finite reflection group where V is r-dimensional. Choose a basis of V so that G is identified with a subgroup of GLr(C). Now, given an automorphism σ of the field C,
applying σ to all entries of the matrices of G defines another group σG
and so another complex finite reflection group (V,σG).
Then it follows from Theorem 2.26 that there exists φ∈ GL(V ) and a ∈ Aut(G) such that, for all g ∈ G,
σ(g) = φa(g)φ−1.
Definition 2.29. A finite reflection group (V, G) is said to be well-generated if G may be well-generated by r reflections, where r = dim(V ).
The well-generated irreducible groups are G(d, 1, r), G(e, e, r) and all the exceptional groups excepted G7, G11, G12, G13, G15, G19, G22, G31.
Field of definition.
The following theorem has been proved (using a case by case analysis) by Benard [Ben] (see also [Bes1]), and generalizes a well known result on Weyl groups.
Theorem–Definition 2.30. Let (V, G) be a finite complex reflection group. Let QGbe the field generated by the traces on V of all elements of
G. Then all irreducible QGG–representations are absolutely irreducible.
The field QG is called the field of definition of the reflection group
(V, G).
• If QG⊆ R, then (V, G) is a (finite) Coxeter group.
• If QG= Q, then (V, G) is a Weyl group.
2.4. Parabolic subgroups.
Throughout this subsection we assume only that V is a k-vector space of finite dimension, and that G is a finite subgroup of GL(V ).
We denote by Ref(G) the set of all reflections of G, and by Arr(G) the set of reflecting hyperplanes of elements of Ref(G).
Notice that, since G is finite and k of characteristic zero, the kG-module V is completely reducible.
Definition 2.31. We denote by ArrX(G) the set of reflecting
hyper-planes containing X, and by FX the flat of X in Arr(G):
FX :=
\
H∈ArrX(G)
H .
The assertion (1) of the following theorem has first been proved by Steinberg [St]. A short proof may now be found in [Le].
(1) The fixator CG(X) of X is generated by those reflections whose
reflecting hyperplane contains X.
(2) The flat FX is the set of fixed points of CG(X) and there exists a
unique CG(X)–stable subspace VX of V such that V = FX⊕VX .
(3) CG(X) = CG(FX) and NG(X)/CG(X) is naturally isomorphic
to a subgroup of GL(FX).
Proof of (2). Since CG(X) is generated by reflections whose reflecting
hyperplanes contain FX, we see that the flat FX is fixed by CG(X).
Conversely, if x∈ V is fixed under CG(X), it is fixed by all the
reflec-tions of CG(X), hence belongs to FX.
If FX = 0, the assertion (2) is obvious. Assume FX 6= 0. Then
CG(X)6= 1. Since FX is the trivial isotypic component of CG(X), the
space VX is the sum of all nontrivial isotypic components.
Definition 2.33. The fixators of subsets of G in V are called parabolic subgroups of G.
By Theorem 2.32above, a parabolic subgroup CG(X) acts faithfully
as an essential reflection group on the uniquely defined subspace VX.
Corollary 2.34. The map F 7→ CG(F ) is an order reversing bijection
from the set of all flats of Arr(G) onto the set of parabolic subgroups of G (where both sets are ordered by inclusion).
2.5. Linear characters of a finite reflection group. Let (V, G) be a finite reflection group.
The following description of the linear characters of a reflection group, inspired by the results of [Co], may be found, for example, in [Bro2, Theorem 3.9].
Denote by Gab the quotient of G by its derived group, so that
Hom(G, C×) = Hom(Gab, C×) . Recall that Arr(G) denote the
collec-tion of reflecting hyperplanes of the refleccollec-tions s for s∈ G.
In what follows, the notation H ∈ Arr(G)/G means that H runs over a complete set of representatives of the orbits of G on the set Arr(G) of its reflecting hyperplanes.
Theorem 2.35.
(1) The restrictions from G to CG(H) define an isomorphism
Hom(G, C×)−→∼ Y
H∈Arr(G)/G
Hom(CG(H), C×) .
(2) The composition iH : CG(H)→ G → Gab is injective, and
Y H∈Arr(G)/G iH : Y H∈Arr(G)/G CG(H)→ Gab is an isomorphism.
Corollary 2.36. Let S be a generating set of reflections for G and let O be the set of G-conjugates of the elements of S. Then for any H ∈ Arr(G) the set O ∩ CG(H) generates CG(H).
Proof. The set O ∩ CG(H) has the same image in Gab as the set SH of
elements of S which are conjugate to an element of CG(H). If we denote
x 7→ xab the quotient map G → Gab, we have Sab = `
H∈Arr(G)/GSHab,
where Sab
H lies in the component CG(H) of Gab. Since S generates G,
Sab generates Gab, thus Sab
H generates CG(H).
Definition 2.37. Let G be a finite subgroup of GL(V ) generated by reflections. A reflection s ∈ G is said to be distinguished with respect to G if det(s) = exp 2πi
d
where d =|CG(Hs)|.
In particular, if H is a reflecting hyperplane for a reflection of G, every CG(H) is generated by a single distinguished reflection.
The next property has been noticed by Nebe ([Ne], §5), as a conse-quence of [Co, (1.8) & (1.9)].
Corollary 2.38. Let S be a generating set of distinguished reflections for G. Then any distinguished reflection of G is conjugate to an element of S.
Proof. It follows from 2.36 and from the fact that the conjugate of a distinguished reflection is still distinguished.
3. Root Systems Notation and conventions.
From now on, the following notation will be in force.
The field k is a number field, stable by the complex conjugation denoted λ 7→ λ∗. Its ring of integers is Z
k, a Dedekind domain. A
fractional ideal is a finitely generated Zk-submodule of k. Denote by
λZk the (principal) fractional ideal generated by λ∈ k.
For a a fractional ideal, we set
a−1 :={b ∈ k | ba ⊂ Zk}, and a−∗ := (a−1)∗.
Since Zk is Dedekind, aa−1 = 1Zk and (λZk)−1 = λ−1Zk for λ∈ k.
Throughout, (V, W ) is a pair of finite dimensional k-vector spaces with a given Hermitian pairing (see Subsection 2.1)
V × W → k : (v, w) 7→ hv, wi .
For I a finitely generated Zk-submodule of V and J a finitely
gen-erated Zk-submodule of W , we denote by hI, Ji the fractional ideal
generated by all hα, βi for α ∈ I and β ∈ J.
Let I be a rank one finitely generated Zk-submodule of V , generating
is a fractional ideal a of Zk such that I = av. If, similarly, J = bw for
some fractional ideal b and some w ∈ kJ, then hI, Ji = ab∗hv, wi . 3.1. Zk-roots.
Definition 3.1.
(1) A Zk-root (for (V, W )) is a triple r = (I, J, ζ) where
• I is a rank one finitely generated Zk-submodule of V ,
• J is a rank one finitely generated Zk-submodule of W ,
• ζ is a nontrivial root of unity in k,
such that hI, Ji = (1 − ζ)Zk, the principal ideal generated by
1− ζ.
A Zk-root r = (I, J, ζ) is called a (ζ, Zk)-root.
(2) If r = (I, J, ζ) is a Zk-root, and a is a fractional ideal, we set
a· r := (aI, a−∗J, ζ) .
Two Zk-roots r1 and r2 are said to be of the same genus if there
exists a fractional ideal a such that r2 = a· r1.
The group GL(V ) acts on left on the set of Zk-roots, as follows: for
g ∈ GL(V ) and r = (I, J, ζ) a Zk-root, set
g· r := (g(I), g∨(J), ζ).
In particular λ ∈ k× ⊂ ZGL(V ) acts by λ · r = (λI, λ−∗J, ζ). The
action of k×Id
V = ZGL(V ) preserves genera.
Remark 3.2. The pair (I, J) does not determine ζ.
Indeed one may have an equality of ideals (1 − ζ)Zk = (1− ξ)Zk
without ζ and ξ having even the same order. For example, as soon as ζ has a composite order, 1− ζ is invertible and so (1 − ζ)Zk= Zk (see
Lemma A.3 in Appendix A).
Given a Zk-root r = (I, J, ζ), choose v ∈ kI and w ∈ kJ such that
hv, wi = 1 − ζ. Then the formula
x7→ x − hx, wiv
defines a reflection independent of the choice of v, since it is also the reflection attached to the reflection triple (kI, kJ, ζ). We will denote by sr this reflection.
Definition 3.3.
(1) If s is a reflection, an (s, Zk)-root is a Zk-root (I, J, ζ) where
(kI, kJ, ζ) = (Ls, Ms, ζs).
(2) If r = (I, J, ζ) is a Zk-root for (V, W ), we call r∨ = (J, I, ζ) —
Notice that the dual of an (s, Zk)-root is an (s∨, Zk)-root. Thus
sr∨ = s∨r.
Lemma 3.4.
(1) Given a Zk-root r = (I, J, ζ), given v ∈ kI and w ∈ kJ such
that hv, wi = 1 − ζ, there exists a fractional ideal a such that I = av and J = a−∗w.
(2) For any Zk-root r = (I, J, ζ), there exists a unique reflection s
in GL(V ) such that r is an (s, Zk)-root.
(3) For any reflection s in GL(V ), the set of (s, Zk)-roots form a
single genus of roots.
Proof. (1) and (2) are clear. Let us prove (3). Let s be a reflection. Choose v ∈ Ls and w ∈ Ms such that hv, wi = 1 − ζs. For a any
fractional ideal, define I := av, J := a−∗w . Then r = (I, J, ζ
s) is an
(s, Zk)-root.
Let now r′ be a root giving rise to the same reflection triple. Then
r′ = (bv, b−∗w, ζ) for some fractional ideal. We have r′ = ba−1· r thus
r and r′ are in the same genus.
Remark 3.5. Given a reflection s and an (s, Zk)-root r = (I, J, ζ),
Lemma 3.4, (1) ensures that J is determined by I (and similarly I is determined by J).
Pairing between Zk-roots.
Let r1 = (I1, J1, ζ1) and r2 = (I2, J2, ζ2) be two Zk-roots. There is a
pairing on the set of Zk-roots, defined to be the fractional ideal:
n(r1, r2) := hI1, J2i .
If r = (I, J, ζ), then by definition we have n(r, r) = (1− ζ)Zk.
Principal Zk-roots.
Let I be a rank one Zk-submodule of V . The reader will easily check
that the following assertions are equivalent: (i) I is a free Zk-module (hence of rank 1),
(ii) whenever v ∈ kI and a is a fractional ideal of k such that I = av, then a is a principal ideal.
This implies the following result:
Lemma–Definition 3.6. Let r = (I, J, ζ) be a Zk-root. The following
assertions are equivalent:
(i) I is a free Zk-module (hence of rank 1),
(ii) J is a free Zk-module (hence of rank 1).
If the preceding properties are true, we say that the root r is a principal Zk-root.
Remark 3.7. If r = (I, J, ζ) is a principal Zk-root, we may choose
α ∈ kI and β ∈ kJ such that I = Zkα, J = Zkβ and hα, βi = 1 − ζ.
The vector α is then unique up to multiplication by a unit of Zk, and
it determines β (and conversely). 3.2. Zk-root systems.
Definition and first properties.
The following definition is modeled on that of Bourbaki [BouLie, chap. VI, §1, D´efinition].
Definition 3.8. Let R ={r = (Ir, Jr, ζr)} be a set of Zk-roots. We say
that R is a Zk-root system if it satisfies the following conditions:
(RSI): R is finite, and the family (Ir)r∈R generates V ,
(RSII): Whenever r∈ R, we have sr· R = R,
(RSIII): Whenever r1, r2 ∈ R, we have n(r1, r2)⊆ Zk.
In particular, in the case when Zk = Z, the root datum above is
equivalent to that required for a root system as defined in loc.cit. (see Remark 3.10).
If G is any of the 34 exceptional reflection groups of the classification of finite irreducible complex reflection groups, and k = QG is the field
of definition of G, then Zk is known to be a principal ideal domain [Ne].
Principal Zk-root systems.
If Zk is a principal ideal domain, all Zk-roots are principal.
Definition 3.9. A Zk-root system is principal if all its roots are
prin-cipal.
Remark3.7implies that a principal Zk-root may be viewed as a triple
(A, B, ζ) where
• ζ is a root of unity,
• A = Z×kα and B = Z×kβ, where α and β are nonzero elements
of V and W respectively, and • hα, βi = 1 − ζ .
Such a triple r defines the unique reflection sr with reflecting line kA
and reflecting hyperplane the orthogonal of kB.
Thus a principal Zk-root system may be viewed as a set R of triples
r = (Ar, Br, ζr)r∈R such that
(RSI) R is finite and the family (Ar)r∈R generates V ,
(RSII) Whenever r∈ R, we have sr· R = R ,
(RSIII) Whenever r1 = (A1, B1, ζ1) ∈ R and r2 = (A2, B2, ζ2) ∈ R, for
Remark 3.10. If Zk = Z (which implies that G is a Weyl group), the
previous definition coincides with the usual definition of root system attached to G: let R0 be a root system in the Bourbaki sense, then
R:={(Zα, Zα∨,−1) | α ∈ R0}
is a Z-root system in our sense. Notice that the cardinality of R0 is
twice that of R as Bourbaki has distinct roots ±α, which give rise to a single Z-root.
Remark 3.11. Nebe’s definition of a reduced k-root system for G (see [Ne, Def.19]) coincides with our definition of distinguished principal Zk-root system for G (see Definition 3.24 below).
Reflections and integrality results.
We return to the general case, where Zk need not be a P.I.D.
Lemma 3.12. Given Zk-roots r1 = (I1, J1, ζ1) and r2 = (I2, J2, ζ2),
(1) (sr1 − IdV)(I2)⊂ n(r2, r1)I1.
(2) If n(r2, r1)⊂ Zk, then (sr1 − IdV)(I2)⊂ I1.
(3) Reciprocally, if (sr1 − IdV)(I2)⊂ I1, then n(r2, r1)⊂ Zk.
Proof. Choose (v1, w1) ∈ kIr1 × kJr1 such that hv1, w1i = 1 − ζ1, and
denote by a1 the fractional ideal such that Ir1 = a1v1 (and so Jr1 =
a−∗1 w1).
Similarly, choose (v2, w2) ∈ kIr2 × kJr2 such that hv2, w2i = 1 − ζ2,
and denote by a2 the fractional ideal such that Ir2 = a2v2 (and so
Jr2 = a−∗2 w2).
Then, for all a2 ∈ a2,
(⋆) sr1(a2v2) = a2v2 − ha2v2, w1iv1.
In order to prove (1), write 1 = Piyixi for xi ∈ a1 and yi ∈ a−11 .
Then the above equality (⋆) may be rewritten sr1(a2v2) = a2v2− ha2v2, w1i( X i yixi)v1 = a2v2− X i ha2v2, yi∗w1ixiv1 ,
and that last equality shows (1).
Part (2) follows from (1) and from the inclusion ZkI1 ⊂ I1.
Now assume that (sr1− IdV)(Ir2)⊂ Ir1. Equality (⋆) shows that, for
all a2 ∈ a2, ha2v2, w1iv1 ∈ I1, i.e., ha2v2, w1i = a2hv2, w1i ∈ a1. This
shows that a2hv2, w1i ⊂ a1, hence hv2, w1i ∈ a−12 a1 andha2v2, a−∗1 w1i ⊂
Zk, which is (3).
Corollary 3.13. Condition (RSIII) is equivalent to:
The group G(R).
Definition 3.14. Given a set R of Zk-roots, we denote by G(R) the
subgroup of GL(V ) generated by the family of reflections (sr)r∈R.
If G(R) = G, we say that R is a Zk-root system with group G, or a
Zk-root system for G.
Remark 3.15. By Definition 3.8, (RSII), g(r) ∈ R whenever g ∈ G(R)
and r ∈ R. Theorem 3.16.
(1) Let R be a Zk-root system in V . Then (V, G(R)) is an essential
finite reflection group – that is, VG(R) = 0.
(2) Let (V, G) be an essential finite reflection group. Then there exists a Zk-root system in V with group G.
Proof. (1) The set of reflections SR={sr | r ∈ R} is finite, by (RSI); is
saturated, by (RSII); and generates G(R), by definition. Thus
Propo-sition 2.17 applies, and G(R) is finite.
(2) If X is a finite spanning set of V , the (finite) set {g(x) | (x ∈ X)(g ∈ G)}
generates a finitely generated Zk-submodule E of V , which generates
V as a k-vector space, and which is G-stable. Since E is torsion free (and since Zk is Dedekind), E is a lattice in V , namely there exists
a family E1, . . . , Er of rank one projective Zk-modules such that E =
E1⊕· · ·⊕Er. Notice that if L is any line in V , then L∩E 6= 0. Indeed,
since kE = V , we have L ⊂ kE, thus for each x ∈ L, x 6= 0, there is m ∈ E and λ, µ ∈ Zk, λµ 6= 0, such that x = λµm, so λm is a nonzero
element of L∩ E.
Let W be a vector space with a Hermitian pairing with V (for ex-ample, the twisted dual ∗V ). For each s∈ Ref(G), with reflecting line
Ls, dual reflecting line Ms, and determinant ζs,
• set Is := Ls∩ E (so Is is a rank one Zk-submodule of E), and
• denote by Js the rank one Zk-submodule of W in Ms such that
hIs, Jsi = (1 − ζs)Zk.
Then (Is, Js, ζs) is an (s, Zk)-root.
Denote by RE the set of all roots (Is, Js, ζs) for s ∈ Ref(G). It is
clear that G(RE) = G. It remains to show that RE is a Zk-root system.
(RSI) : It is clear that RE is finite. Besides, since VG= 0, we have
V =Ps∈Ref(G)Ls, hence the family (Is)s∈Ref(G) generates V .
(RSII) : Let g ∈ G. Whenever s ∈ Ref(G), we have g(Ls) = Lgs,
hence g(Ls∩ E) = Lgs∩ E, which shows that g(Is) = Igs. It follows
(RSIII) : Let s1, s2 ∈ Ref(G), and set ri := (Ii, Ji, ζi) for i = 1, 2.
Since the image of s2− Idv is the line Ls2, and since s2− IdV sends
E to E, we see that
(s2− IdV)(I1)⊂ E ∩ Ls2 = I2.
This condition is equivalent to (RSIII) by Corollary 3.13.
Remark 3.17. If r1 and r2 belong to the same root system R and
n(r1, r2) = 0 then sr1sr2 = sr2sr1 and n(r2, r1) = 0. Indeed, by item
(1) of Proposition 2.23, which is applicable since G(R) is finite, the equality n(r1, r2) = 0 implies that the reflection triples defined by r1
and r2 are orthogonal.
Let (V, G) be a finite reflection group on k, assumed to be essential (Definition 2.10). Let S be a set of reflections of G, and for each s ∈ S let rs = (Is, Js, ζs) be a (s, Zk)-root (see Definitions 3.3). We set
S := {rs | s ∈ S}.
A root r is said to be distinguished with respect to a Zk-root system
R if sr is distinguished with respect to G(R).
Proposition 3.18. Assume that (a) the set S generates G,
(b) for each s, t∈ S, the ideal hIs, Jti is integral.
Then
(1) the orbit R of S under G is a Zk-root system, and G = G(R),
(2) if all elements of S are distinguished, then so are the elements of R, and the map R→ Ref(G), r 7→ sr is a bijection onto the
set of distinguished reflections of G, and
(3) if each element of S is principal, then R is principal as well. Proof. (1) The axiom (RSI) follows from the fact that (V, G) is
essen-tial, and the axiom (RSII) is trivial. Let us prove (RSIII).
Lemma 3.19. Let s, t, u be reflections on V , with associated Zk-roots
respectively rs = (Is, Js, ζs), rt, ru. Then
hIsts−1, Jui ⊂ hIs, Jui + hIt, JsihIs, Jui .
Proof of 3.19. By item (1) of Lemma 3.12, s(It)⊂ It+hIt, JsiIs hence
hIsts−1, Jui ⊂ hIs, Jui + hIt, JsihIs, Jui .
We return to the proof of Proposition 3.18. Say that a set T of re-flections of G is integral if s, t ∈ T implies hIs, Jti ∈ Zk. The preceding
lemma shows that, given any integral set T of reflections of G, then T ∪ {sts−1 | s, t ∈ S} is again an integral set of reflections of G. Thus
the set of all conjugates of the elements of S is integral, which is axiom (RSIII).
(2) Since the elements ofS are all distinguished, the assertion results from Corollary 2.36.
Case where V = W and h-, -i is positive.
Remark 3.20. If V = W and h-, -i is positive, a reflection s ∈ G is determined by its root line and its determinant. Indeed, since s pre-serves h-, -i, we have s = s∨, so if s is associated to the reflection triple
(L, M, ζ), we have L = M.
Lemma 3.21. Assume V = W and h-, -i is positive.
(1) Let r = (I, J, ζ) be a Zk-root. Then J = (1− ζ∗)hI, Ii−1I.
(2) Let r, r′ be two roots from a Z
k-root system R. Then
n(r′, r)∗ = n(r, r)∗n(r′, r′)−1n(r′, r′∨)n(r, r∨)−1n(r, r′). Proof.
(1) Choose v ∈ kI, w ∈ kJ such that hv, wi = 1 − ζ. There is a fractional ideal a such that I = av and J = a−∗w. By Remark 3.20
there exists λ ∈ k such that w = λv thus 1 − ζ = λ∗hv, vi and w = (1−ζ∗) hv,vi v. Thus J = a−∗w = (1− ζ∗) v a∗hv, vi = (1− ζ ∗) av aa∗hv, vi = (1− ζ ∗) I hI, Ii. Part (2) follows from (1) observing that for a root r = (I, J, ζ) we have n(r, r) = (1− ζ)Zk and n(r, r∨) =hI, Ii.
Remark 3.22. In the case where k ⊂ R, and r = (I, J, −1), r′ =
(I′, J′,−1), Lemma 3.21 (2) reduces to n(r′, r) = hI′, I′ihI, Ii−1n(r, r′)
which generalizes the case of finite Coxeter groups [BouLie, Chap 6, §1, no. 1.1. formula (9)].
Some properties of a root system.
We return to the general case, where W need not be the same as V . Let R be a Zk-root system. Recall that G(R) (or simply G) denotes
the group generated by the reflections defined by the elements of R. Proposition 3.23. Let R be a Zk-root system. Then, for any reflecting
hyperplane H of G(R), the fixator of H, CG(R)(H), is generated by the
set of reflections sr (where r∈ R) with reflecting hyperplane H.
Proof. It is a consequence of Corollary2.36. Indeed, it suffices to notice (see Remark 3.15) that, for r ∈ R and g ∈ G(R), then gsrg−1 = sg(r)
and g(r)∈ R.
Definition 3.24. Let R be a Zk-root system.
(1) We say that R is reduced if the map r7→ sr is injective.
(2) We say that R is complete if the map r7→ sr is surjective onto
Ref(G(R)).
(3) We say that R is distinguished if
(a) it consists of distinguished roots, and (b) it is reduced.
Remarks 3.25.
(1) If all srhave order 2 (for example, the real reflection groups and
the infinite family G(e, e, r)) then:
• every distinguished root system is complete (and reduced), and
• every complete and reduced root system is distinguished. (2) In a reduced root system, distinct roots have different genus. Proposition 3.26. Let R be a distinguished Zk-root system. Then
the map r 7→ sr is a bijection from R onto the set of distinguished
reflections of G(R).
Proof. It suffices to prove that, whenever H is a reflecting hyperplane of G(R), there exists r ∈ R such that sr is the distinguished reflection
of CG(R)(H). This results from 3.23.
The following lemma follows the lines of [Ne, Remark 20].
Proposition 3.27. Let (V, G) be a k-reflection group. Let R be a re-duced and complete Zk-root system (resp. a distinguished Zk-root
sys-tem) with respect to G. Assume that R1, . . . , Rm are the orbits of G
on R.
Then any other reduced and complete Zk-root system (resp.
distin-guished Zk-root system) with group G is of the form a1R1∪ · · · ∪ amRm
where a1, . . . , am are some fractional ideals.
Proof. We give the proof for reduced and complete systems (the proof for the distinguished ones is similar).
Let R′ be another reduced and complete root system with respect
to G. If s is a reflection of G, and if rs ∈ R and r′s ∈ R′ are the roots
associated with s, we know by Lemma 3.4that there exists a fractional ideal as 6= 0 such that r′s = as· rs.
Choose i such that 1 ≤ i ≤ m and choose a reflection s such that rs ∈ Ri. For g ∈ G, g(rs) is a Zk-root attached to g(s), hence by
hypothesis we have g(rs) = rgsg−1. Similarly, g(r′
s) = r′gsg−1. Hence
we see that as = ag(s), thus as depends only on i. We set ai := as.
This shows that if R1, . . . , Rm are the orbits of G on R, then for each
i = 1, . . . , m, there exists a fractional ideal aisuch that a1R1, . . . , amRm
are the orbits of G on R′.
Distinguishing and completing root systems.
Proposition 3.28. Any Zk-root system R contains a reduced
subsys-tem with group G(R).
Proof. Let [Arr(G(R))/G(R)] be a complete set of representatives of orbits of G(R) on the set Arr(G(R)) of reflecting hyperplanes of G(R). For each H ∈ [Arr(G(R))/G(R)], let us set
By Corollary 2.36, we know that RefR(H) generates CG(R)(H). For
each H ∈ [Arr(G(R))/G(R)], choose a subset Ref0R(H) of RefR(H)
which is minimal subject to being a generating subset of CG(R)(H).
For each element of Ref0R(H), we choose a corresponding r∈ R, so we
get a set {r1, . . . , rn} of elements of R.
Define R′ to be the union of the G(R)-orbits of {r1, . . . , rn}. We
claim that R′ is a reduced root system with group G(R).
It is clear that R′ is a root system with group G(R). Let us prove
that R′is reduced. Suppose that r and g·r = a·r are in R′ for g∈ G(R)
for some fractional ideal a, then gn· r = an· r = r for some n which
shows that an = Z
k which implies (since Zk is a Dedekind domain,
hence fractional ideals have a unique decomposition in prime ideals)
that a = Zk.
Proposition 3.29. Let R be a complete Zk-root system. Then R
con-tains a distinguished subsystem with group G(R).
Proof. Denote by R0 the set of distinguished roots of R. Since R is
complete, any distinguished reflection of G(R) is of the form sr for
r∈ R0.
Thus the set of reflecting lines of R0is the same as the set of reflecting
lines of R, and this proves that condition (RSI) (see Definition3.8) is
satisfied for R0.
Condition (RSIII) on R0 is inherited from R.
It remains to check that R0 is stable under G(R0). Notice that
G(R0) = G(R). Thus (RSII) follows from the fact that the image of a
distinguished root by an element of G(R0) is still distinguished.
Now apply Proposition 3.28 to get a reduced subsystem of R0, and
we get a distinguished root system.
Let r = (Ir, Jr, ζ) be a Zk-root with ζ = exp(2πid ) with d > 2. For
1≤ i < d, we denote by ri the root:
ri := 1− ζi 1− ζIr, Jr, ζ i ,
which has the property that si r = sri.
An incomplete root system can be augmented by adjoining all of the ri to obtain a complete root system.
Proposition 3.30. Let R be a distinguished Zk-root system for V .
Denote by bR the set of roots obtained by adjoining all roots of the form ri to R. Then bRis a complete reduced Zk-root system for V , whose set
of distinguished roots is R.
Proof. Since R is finite and only a finite number of roots are added to form bR, the condition (RSI) is immediately satisfied for bR.
Let us check (RSII). Since all the reflections s′rfor r′ ∈ bRbelong to
G(R), it suffices to check that bRis stable under all sr for r∈ R. This
results from the fact that if s(r1) = r2 then s(ri1) = ri2.
Finally, consider two roots ri 1 and r
j
2 in bR, where r1 and r2 are in R.
Then n(ri1, rj2) = 1− ζi 1 1− ζ1 Ir1, Jr2 = 1− ζi 1 1− ζ1 hIr1, Jr2i .
Since n(r1, r2)∈ Zk by (RSIII) for R, and since 1−ζ i 1 1−ζ1 is always integral, n(ri 1, r j
2)∈ Zk as well, verifying the last condition (RSIII) for bR.
The system bRis reduced since sri = sir.
Dual root system, irreducible root systems.
Recall (see Definitions 3.3(2)) that if r = (I, J, ζ) is a Zk-root in V ,
its dual root is the Zk-root in W defined by r∨ := (J, I, ζ).
Lemma–Definition 3.31. If R is a Zk-root system in V , the set
R∨ :={r∨ | r ∈ R}
is a Zk-root system in W , called the dual root system of R.
Proof. The fact that R∨ is a Z
k-root system follows directly from the
definition and the equality:
n(r∨1, r∨2) =hJ1, I2i = hI2, J1i∗ = n(r2, r1)∗.
Recall (Definition 2.12(2)) that a set of reflections is irreducible if it consists of a single equivalence class with respect to the closure of the “is not orthogonal” relation ∼.
Definition 3.32. Let R be a set of Zk-roots, and SR be the
corre-sponding set of reflections. Then R is said to be irreducible if SR is
irreducible.
So a set of roots R is irreducible if for every pair of roots r and r′,
there is a sequence r = ri0, ri1, . . . , rip = r′ such that each adjacent pair
of roots in the sequence is not orthogonal – that is, n(rij, rij+1)6= 0.
3.3. Root systems and parabolic subgroups.
In this subsection, R denotes a Zk-root system, and G := G(R).
Let F be a flat of G in V , that is, an intersection of reflecting hyper-planes of G in V . Recall that we denote by ArrF(G) the family of all
reflecting hyperplanes of G containing F , so that F =TH∈ArrF(G)H . For H ∈ ArrF(G), we denote by LH the reflecting line in V attached
to H (see Proposition 2.22), by MH the orthogonal of H in W (a dual
reflecting line for G in W ), and by KH the corresponding dual reflecting
We set VF := X H∈ArrF(G) LH and WF := X H∈ArrF(G) MH.
The Hermitian pairing between V and W restricts to a Hermitian pair-ing between VF and WF.
Let CG(F ) be the corresponding parabolic subgroup of G, the fixator
of F . We recall (see Theorem 2.32 above) that
• CG(F ) is generated by those reflections whose reflecting
hyper-planes belong to ArrF(G),
• F is the set of fixed points of CG(F ) in V ,
and CG(F ) is naturally identified with a subgroup of GL(VF) generated
by reflections.
Let r = (I, J, ζ) ∈ R such that sr ∈ CG(F ). Then sr is a reflection in
its action on VF, and r may be viewed as a Zk-root for (VF, WF), since
I ⊂ VF and J ⊂ WF.
Proposition 3.33. Let R be a Zk-root system in V , and let F be a
flat of G(R) in V . (1) The set
RF :={r | sr∈ CG(F )}
is a Zk-root system for the parabolic subgroup CG(F ) viewed as
a reflection group acting on VF.
(2) If R is complete then RF is a complete root system for CG(F ).
(3) If R is distinguished then RF is a distinguished root system for
CG(F ).
Proof. (1) It suffices to check that RF is stable under the action of
CG(F ). It is enough to check that for r∈ RF and t ∈ CG(F ), we have
t· r ∈ RF. But st·r= tsrt−1, which fixes F .
Completeness and distinguishedness (items (2) and (3)) are inherited
directly from R.
3.4. Root lattices, root bases.
Definition 3.34. Let R ={ r = (Ir, Jr, ζr)} be a Zk-root system.
(1) The root lattice QR and the coroot lattice Q∨R are defined by
QR := X r∈R Ir and Q∨R := X r∈R Jr.
(2) The weight lattice PR and the coweight lattice PR∨ are the dual
of Q∨
R and QR respectively, i.e.,
PR :={ x ∈ V | hx, Q∨Ri ⊆ Zk}
PR∨ :={ y ∈ W | hy, QRi ⊆ Zk}
The following properties are straightforward. • QR ⊆ PR and Q∨R ⊆ PR∨.
• QR∨ = Q∨R, PR∨ = PR∨.
Definition 3.35. The group of automorphisms of a Zk-root system R
denoted Aut(R), is the group of all g ∈ GL(V ) such that g(R) = R. In other words,
Aut(R) ={g ∈ GL(V ) | (I, J, ζ) ∈ R ⇒ (g(I), g∨(J), ζ)∈ R} .
If g ∈ Aut(R), g conjugates the reflection sr defined by a root r∈ R
to the reflection sg(r) defined by g(r), hence
G(R) ⊳ Aut(R) . Proposition 3.36.
(1) The lattices QR, PR, Q∨R, PR∨ are all Aut(R)-stable finitely
gen-erated projective Zk-submodules of V and W respectively.
(2) The group G(R) acts trivially on PR/QR and on PR∨/Q∨R, hence
the group Aut(R)/G(R) acts on these quotients.
(3) The Hermitian pairing V × W → k induces a non-degenerate pairing of Zk(Aut(R)/G(R))–modules
PR/QR× PR∨/Q∨R −→ k/Zk.
Proof. Assertion (1) is clear. Assertion (2) results from the following lemma, which gives an alternative description of the reflection associ-ated with a Zk-root.
Lemma 3.37. Let r = (I, J, ζ) be a Zk-root. Assume that (αi)i∈E and
(βi)i∈E are finite families of elements of I and J respectively such that
X
i∈E
hαi, βii = 1 − ζ .
Then, for all v ∈ V ,
sr(v) = v−
X
i∈E
hv, βiiαi.
Proof. For all i ∈ E, we have (see Proposition 2.4) sr(v) = v− hv, β
ii
hαi, βii
(1− ζ)αi, hence
hαi, βiisr(v) =hαi, βiiv − hv, βii(1 − ζ)αi.
Summing the last equality over E, then simplifying by (1− ζ), gives
the expected formula.
Now (with the same notation as in the above lemma), for all v ∈ PR,
hv, βii ∈ Zk, hence hv, βiiαi ∈ QR, which shows that sr acts trivially
on PR/QR.
Notice also that for a a fractional ideal, we have Qa·R = aQR , Q∨a·R = a−∗Q∨R
Pa·R = aPR , Pa∨·R = a−∗PR∨.
Definition 3.38. Let R1 and R2 be two Zk-root systems.
(1) Say that R1 and R2 are of the same genus if there exists a
fractional ideal a such that
R2 = a· R1 :={ a · r1 | (r1∈ R1)}.
(2) Say that R and R′ are lattice equivalent if there exists a
frac-tional ideal a such that QR′ = aQR and Q∨R′ = a−∗Q∨R.
Notice that if two root systems have the same genus, then they are lattice equivalent. The converse is false, as seen in the following exam-ple.
Example 3.39. Consider k := Q(ζ3), hence Zk = Z[ζ3] (a principal ideal
domain). Set W = V = k and G := µ3 acting on V by multiplication. Now (1− ζ2
3)Zk = (1− ζ3)Zk, since 1− ζ32=−ζ(1 − ζ3).
It is easily checked (see Section 4 below for the general case of the cyclic groups) that there are exactly three genera of complete root systems for G as follows: let p = (1− ζ3)Zk = (1− ζ32)Zk, then
R1,1 := n Zk, p, ζ3 , Zk, p, ζ32 o R1,p := n Zk, p, ζ3 , p, Zk, ζ32 o Rp,1 := n p, Zk, ζ3 , Zk, p, ζ32 o
Now QR1,p = QRp,1 = Zk, and PR1,p = PRp,1 = Zk, so these two distinct
(with respect to genus) root systems are in fact lattice equivalent. Definition 3.40. A subset Π of elements of R is said to be
• a set of root generators if QR =Pr∈ΠIr
• a root lattice basis if QR =Lr∈ΠIr
• a root basis if
(1) QR=Lr∈ΠIr and
(2) the family (sr)r∈Π generates G(R).
Coroot generators, coroot lattice bases, coroot bases are defined anal-ogously.
Example 3.41. As above, let us choose k := Q(ζ3), hence Zk = Z[ζ3],
V = W = k and G := µ3 acting on V by multiplication. Then each of the root systems R1,1, R1,p, Rp,1 contains a root basis – for example,
Π = nZk, (1− ζ3)Zk, ζ3
o
. Indeed, Π is both a root basis and a coroot basis of R1,1. However R1,p and Rp,1 do not contain a subset
On the other hand, a distinguished root system for a well generated group always contains a subset which is simultaneously a root basis and a coroot basis:
Proposition 3.42. Let R be a distinguished Zk-root system. Let Π be
a subset of R such that {sr | r ∈ Π} generates G(R). Then
(1) whenever r ∈ R, there exist r0, r1, . . . , rm ∈ Π such that r =
(srm· · · sr1)· r0,
(2) Π is a set of root generators and a set of coroot generators, (3) if Π consists of principal Zk-roots, then R is principal,
(4) if |Π| = dim V , then Π is a root basis and a coroot basis
Proof. Write G for G(R). Notice that it is enough to prove the results concerning roots: the ones concerning coroots follow by considering the contragredient operation g 7→ g∨ of G on W .
(1) Let r ∈ R. By Corollary2.38, sr is conjugate to some sr0 for r0 ∈
Π. Thus there exist r1, . . . , rm ∈ Π with sr = srm· · · sr1sr0s−1r1 · · · s−1rm .
Since R is stable under G, and since wsr0w−1 = sw.sr0, the above
equal-ity implies that r = (srm· · · sr1)· r0.
(2) Suppose that for r, r0, . . . , rm ∈ Π are as in (1). By Lemma3.12,
sr1(Ir0)⊂ Ir0 + Ir1,
so
sr2sr1(Ir0)⊂ sr2(Ir0) + sr2(Ir1)⊂ Ir0 + Ir1 + Ir2,
and an iteration shows that
Ir= (srm· · · sr1)(Ir0)⊂ Ir0 + Ir1 +· · · + Irm.
Part (3) results from assertion (1) and from the remark that, for any g ∈ GL(V ) and r a principal Zk-root, g(r) is still principal.
Item (4) is clear.
Proposition 3.43. Let R be a distinguished Zk-root system. If Π is
a subset of R such that |Π| = dim V and the family (sr)r∈Π generates
G(R), then Π is a root basis and coroot basis of R.
Proof. This results from Proposition 3.42 (4).
Note that root bases only exist when G(R) is well generated. Remark 3.44. When Zkis not a P.I.D., a root basis does not necessarily
provide a basis of QR as a Zk-module. Nevertheless, we shall see later
(Theorem 6.6, see also [Ne, Corollary 13]) that every reflection group has at least one principal Zk-root system, and the root lattice of a
3.5. Example: the Weyl group of type B2.
Let k be a number field. Set V = k2with canonical basis{e
1, e2} and
W = k2 with canonical dual basis{f
1, f2}. The Weyl group of type B2,
denoted G, may be considered to be the subgroup of GL(V ) generated by S = {s, t} where s and t are the automorphisms of V corresponding respectively to the following matrices on the basis {e1, e2}:
s := −1 0 0 1 and t := 0 1 1 0 .
The corresponding reflecting lines are
Ls = kvs with vs= e1 and Lt= kvt with vt= e2− e1,
Ms = kv∨s with vs∨ = 2f1 and Mt = kvt∨ with vt∨ = f2− f1.
The orbits under G of the following root bases (corresponding to generators s and t in that order) are Zk-root systems corresponding to
the types B2 and C2 respectively:
Π(B2) := n Zkvs, Zkvs∨,−1 , Zkvt, Zkv∨t,−1 o , Π(C2) := 2Zkvs, 1 2Zkv ∨ s,−1 , Zkvt, Zkv∨t,−1 .
Swapping V and W , and s and t, defines an isomorphism between the coroot system of type B2 and the root system of type C2, and vice
versa. We say that they are mutually dual root systems.
It is immediate to check that the element φ∈ GL(V ) defined by φ :
e
1 7→ −e1+ e2,
e2 7→ e1+ e2,
that is, the automorphism of V with matrix
−1 1
1 1
on the basis (e1, e2), has the following properties:
(1) φ2 = 2Id V ,
(2) it swaps s and t (by conjugation),
(3) it sends Π(B2) onto Π(C2) and Π(C2) onto 2Π(B2), hence swaps
R(B2) and R(C2), up to genus.
That is, the automorphism denoted 2B
2 of G swaps, up to genus, R(B2)
and R(C2), which are thus isomorphic (up to genus).
Lemma 3.45.
(1) The following assertions are equivalent.
(i) There exists a Zk-root system with group G which is stable
by the automorphism φ (up to genus),
(2) If a = (Zka) is such that a2 = 2u with u∈ Z×k, we set: Πa:= n aZkvs, a−∗Zkvs∨,−1 , Zkvt, Zkvt∨,−1 o ,
and denote by Ra the orbit of Πa under the group generated by
the reflections sr for r∈ Πa. Then
(a) Ra is a Zk-root system with group G,
(b) the flips between V and W and between s and t define an isomorphism between Ra and its coroot system (thus, Ra
is “self-dual”),
(c) φ(Ra) = aRa (thus Ra is stable by φ up to genus).
That is, the root system Ra affords an automorphism corresponding
to the automorphism 2B
2 of G.
Proof. The proof of (2) is easy and left to the reader. Moreover, (2) implies the implication (ii)⇒(i) of (1).
Let us prove (1), (i)⇒(ii). We may assume that R, the Zk-root
system for G which is stable (up to genus) under φ, has root basis: Π :=n asvs, a−∗s vs∨,−1
, atvt, a−∗t v∨t,−1
o
for some fractional ideals as, at. Now φ(R) = aR for some a ∈ k×, so
asvt= aatvt and 2atvs = aasvs,
from which we deduce that a2at = aas = 2at. Multiplication by a−1t
gives a2Z
k= 2Zk.
For example, set k = Q(i) or Q(√2). The ring Zkis a principal ideal
domain. Setting a := Z[i](1+i) or Z[√2]√2, then a = a∗and 2Z k = a2
is the decomposition of 2Zk in Zk.
It is immediate to check that, if 2Zk = a2, there are at least three
genera of reduced Zk-root systems for G described as the orbits under
G of the following three pairs of roots:
{ (Zkvs, Zkvs∨,−1), (Zkvt, Zkvt∨,−1)}
{ (2Zkvs,12Zkvs∨,−1), (Zkvt, Zkv∨t,−1)}
{ (avs, a−∗vs∨,−1), (Zkvt, Zkv∨t,−1)}.
3.6. Connection index.
Let R = {r = (Ir, Jr, ζr)} be a Zk-root system. The characteristic
ideal (see for example [Bro3, 2.3.4.2]) of the torsion Zk-module PR/QR
is defined by r ^ QR = Ch(PR/QR) r ^ PR,
where r := dim V (see [BouAlg, §4, n06]).
Remark 3.46. The characteristic ideal is the image in the group of fractional ideals of Zk of the divisor called “contenu” in [BouAlg, §4,
The next definition is inspired by the definition given in [BouLie, chap. 6, no 1.9].
Definition 3.47. The characteristic ideal of the torsion Zk-module
PR/QR is called the connection index of the root system R.
Theorem–Definition 3.48. Let (V, G) be an irreducible well gener-ated reflection group. The connection index of a distinguished Zk-root
system R for G does not depend on the choice of R, and is called the connection index of (V, G).
Proof of Theorem 3.48. Let r := dim V . Let R be a distinguished Zk
-root system for G. By item (4) of Proposition 3.42, since (V, G) is well generated, by Proposition3.43 there exists a set Π of r roots such that QR =Lr∈ΠIr and Q∨R =
L
r∈ΠJr.
For all r∈ Π write Ir= arvr and Jr = a−∗r wr for some vectors vr and
wr with hvr, wri = 1 − ζr and some fractional ideal ar. Let w′r be the
dual basis of wr and set Jr′ = arwr′. Then PR =Lr∈ΠJr′.
Assume given another distinguished Zk-root system R′ associated
with the same set of reflections. For each r ∈ R, associated with the reflection sr, let us denote by r′ the element of R′ associated with the
same reflection sr. Then, if r′ = (Ir′, Jr′, ζr) , we have Ir′ = brIr and
Jr′ = b−∗
r Jr for a fractional ideal br.
Then QR′ = M r∈Π brIr and PR′ = M r∈Π brJr′, and r ^ QR′ = Y r∈Π br ^r QR, r ^ PR′ = Y r∈Π br ^r PR.
This shows that
Ch(PR′/QR′) = Ch(PR/QR) ,
and ends the proof.
Remark 3.49. Let V and W be as above, such that dim V = r. Let (V, G) be a reflection group and let s1, . . . , srbe a set of reflections such
that V =Lri=1Lsi and W =Lri=1Msi. For each i = 1, . . . r, pick vi ∈
Lsi and v∨i ∈ Msi such that hvi, v∨ii = 1 − ζsi. Then the Cartan matrix
(hvi, vi∨i)i,j depends only (up to conjugation by a diagonal matrix) on
the choice of the set s1, . . . , sr, hence its determinant depends only on
such a choice.
We shall see later (Proposition 6.7) that if moreover s1, . . . , sr
gener-ate G, that determinant genergener-ates (as an ideal) the connection index, hence in particular it does not depend (up to a unit) on the choice of the generators s1, . . . , sr.
